Extreme points of unit ball in tensor product of spaces Let  $B_1, B_2$ be  unit balls in finite-dimensional normed  spaces $X_1, X_2$ respectively.
Let $e(B_1), e(B_2)$ be corresponding extreme points sets.
Consider the unit ball $B$  in tensor product $X_1\otimes X_2$ with the largest (projective) cross-norm on it. 

Can we say that extreme points of $B$ in tensor product are exactly tensor products of extreme points for $B_1, B_2$, i.e. $e(B)=\{u\otimes w: u\in e(B_1), w\in e(B_2)\}$?     

This seems plausible, but things are not looking very straightforward. In particular, opposit pairs of  extreme points produce the same point in tensor product, i.e. $(-u)\otimes (-w) = u\otimes w$.   
 A: See
[11] Ruess, W.M. and Stegall, C.P., Extreme points in duals of operator spaces, Math. Ann., 261 (1982), 535–546. 
They prove what you want in a more general context:  If $X$, $Y$ are Banach spaces s.t. either $X^*$ or $Y^*$ has the approximation property and either $X^*$ or $Y^*$ has the Radon-Nikodyn property, then the extreme points of the unit ball of the projective tensor product of $X^*$ and $Y^*$ are the tensor products of extreme points of the respective unit balls.
A: Let me add another answer from the perspective of linear algebra/convex geometry. I think the real issue here is that the projective closed unit ball in the algebraic tensor product $E \mathbin{\otimes} F$ is, in general, not the convex hull of $B_E \mathbin{\otimes} B_F$, but rather its closed convex hull. If we omit the closure, even higher-dimensional faces are preserved:

Theorem. ([Dob20, Theorem 3.25]¹) Let $E$ and $F$ be real vector spaces, let $C \subseteq E$, $D \subseteq F$ be symmetric convex sets, and let $M \subset C$, $N \subset D$ be proper faces. Then $\text{conv}(M \mathbin{\otimes} N)$ is a face of $\text{conv}(C \mathbin{\otimes} D)$.
¹: Full disclosure: I am advertising my own paper here. The preceding theorem might have been known before, but I have not been able to find an earlier reference for this result.

In finite-dimensional spaces, $\text{conv}(B_E \mathbin{\otimes} B_F)$ is already closed (by a compactness argument), so here it follows that the projective closed unit ball preserves proper faces. (For extreme points, this also follows from Bill Johnson's answer.)
In general, the preceding theorem does not say anything about the projective unit ball, since an extreme point of $\text{conv}(B_E \mathbin{\otimes} B_F)$ is not necessarily an extreme point of its closure $\overline{\text{conv}}(B_E \mathbin{\otimes} B_F)$.
The other answers suggest that stronger assumptions are needed to ensure preservation of extreme points. I would still be interested in a counterexample to your question, but I guess Bill Johnson's answer suffices for most practical purposes.
References.
[Dob20]: J. van Dobben de Bruyn, Tensor products of convex cones, part I: mapping properties, faces, and semisimplicity, preprint (2020), arXiv:2009.11836v1.
A: There is a hierarchy of notions related to the notion of extreme point, among them strong extreme point, point of continuity, denting point, as sketched in 
BOR-LUH LIN, PEI-KEE  LIN AND S. L. TROYANSKI. CHARACTERIZATIONS  OF  DENTING  POINTS. PROC. OF AMS. Volume  102,  Number  3,  March  1988, p.526-528 (http://www.ams.org/journals/proc/1988-102-03/S0002-9939-1988-0928972-1/S0002-9939-1988-0928972-1.pdf).
The corresponding statement is true for denting points as shown in 
DIRK WERNER. DENTING  POINTS  IN  TENSOR  PRODUCTS OF  BANACH  SPACES. PROC. OF AMS. Volume  101,  Number  1,  September  1987, p. 122-126 (http://www.ams.org/journals/proc/1987-101-01/S0002-9939-1987-0897081-1/S0002-9939-1987-0897081-1.pdf).
Now, it's enough to note that notions of extreme point and denting point are equivalent in my context.
